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In this paper, we generalize the normalized gradient flow method to computethe ground states of Bose-Einstein condensates (BEC) with higher orderinteractions (HOI), which is modelled via the modified Gross-Pitaevskiiequation (MGPE). Schemes constructed in naive ways suffer from severe stabilityproblems due to the high restrictions on time steps. To build an efficient andstable scheme, we split the HOI term into two parts with each part treatedseparately. The part corresponding to a repulsive/positive energy is treatedsemi-implicitly while the one corresponding to an attractive/negative energy istreated fully explicitly. Based on the splitting, we construct theBEFD-splitting and BESP-splitting schemes. A variety of numerical experimentsshows that the splitting will improve the stability of the schemessignificantly. Besides, we will show that the methods can be applied tomultidimensional problems and to the computation of the first excited state aswell.

journal of computational physics

A normalized gradient flow method with attractive-repulsive splitting  for computing ground states of Bose-Einstein condensates with higher-order  interaction